I tutored my dd for about 5 months and got her from a score of 60% in her first full-length practice test to a score of over 90% in about 3 months - so you can do it easily in a year.
And my dd was nearly a year older when she started, in May of Year 5 (exams done then in Nov of year 6). So if your 9-year-old is already getting 50% before the start of year 5, he's in with a good chance.
How to DIY tutor maths for 11+:
- Identify what exams he's actually going to be doing. Is it just maths? Or also VR or English? Which exam boards? Multiple choice or standard (ie writing the answers) format? How long and how many questions? The 11+ forum will tell you all this. No point preparing him for the wrong exam or type of exam. All 11+ maths exams for state schools must limit questions to the content of the KS2 maths syllabus and cannot go beyond it, so any book or website on KS2 maths will be able to tell you what the full list of possible topics are. Remember - these are only aimed at 10-year-olds so are not actually terribly difficult for any normally numerate adult. Plenty of textbooks will explain the topics for you, so don't panic if the last time you studied maths was at school!
- Take the maths paper he did and go through it. The score tells you nothing useful. Discussing the answers will. For every question he got wrong, ask him why he got it wrong. Was it:
a. because he hasn't learnt the topic yet at school? - if so, add it to your list of topics to cover (though bear in mind he'll probably be covering most of them this year at school anyway).
b. because he didn't read the question properly - ie right answer to wrong question. Teach exam technique and focus on learning to read the question properly and identify key information.
c. because he ran out of time? Not a problem at this stage, but get him to swap pens at the end of the 'official' time, so you can see how much he finished in the correct time, and aim to gradually increase the number of questions finished in the timeslot over the year.
d. because he made careless errors? This was my dd's bugbear. I ended up bribing her (successfully) to reduce the number of careless errors to no more than 2 per paper. I was shocked by how much her school let her get away with, in terms of messy layout and lack of attention to detail. The main quality of those who succeed at the 11+ is not brilliance but accuracy.
e. because it's just too hard? - even though they know the topic, had enough time, read the question correctly and made no careless errors? That's actually fine. You'd expect there to be at least 2 or 3 really tough questions per paper, to stretch and identify the most able mathematicians. He doesn't even NEED to get those right - he can get over 90% in the real exam WITHOUT being able to handle those. So your ds shouldn't be put off by those or imagine that all errors fall into this category - they are almost certainly due to factors a. to d. instead. You need to teach him not to stress over these types of questions (let alone for 15 minutes) but to leave them to the end and come back to them if time = basic exam technique.
You should then end up with a list of topics to cover, plus some figures for numbers of careless errors and not-reading-the-questions. That will give you a very clear idea of what topics/skills you need to focus on.
Aim to have covered all the new subject content several months early, so it can sink in. If you have 10 topics and 10 months, say, you can have a very relaxing time of covering one a month. Make sure you start with the basic core maths skills, though - focus on accuracy and speed of basic addition, subtraction, multiplication and division (including fractions and decimals) before you move on to anything else, as those skills will underlie all the other topics. Good mental maths and times tables are essential - you can strengthen these in fun ways like adding up scores on board games or playing card games etc.
Do occasional practice papers so he gets used to them but not loads and certainly not more than 1 a week (arguably none till he's covered the syllabus). Bond do lots of books of them in different formats. As you get nearer the exam, you can start to do more practice papers in the actual style of the one he'll do in the test.
No need to do more than 1 a week - remember, he won't learn anything from doing the test as such, but from going over the answers and understanding how/why he got it wrong, which takes time.
Your 7-year-old shouldn't be doing any of this!! He should be reading as much as possible, learning his times tables and mental maths and playing fun games that improve maths and vocabulary - Boggle, cards, board games etc.
There are starter 11+ books for younger ages, that introduce younger kids to the concepts, in short 10-minute bursts. Maybe he can have a go at one of these books when he feels like it, but that's more than enough. No need at all for him to be doing full, 'proper' 11+ tests at this age, and the results cannot be meaningful in any way, so don't even attempt to analyse them! Let him have fun and enjoy maths and reading instead!